A saturated calculus for the so-called Horn-like sequents of a complete class of a linear temporal logic of the first order is described. The saturated calculus contains neither induction-like postulates nor cut-like rules. Instead of induction-like postulates the saturated calculus contains a finite set of “saturated” sequents, which (1) capture and reflect the periodic structure of inductive reasoning (i.e., a reasoning which applies inductionlike postulates); (2) show that “almost nothing new” can be obtained by continuing the process of derivation of a given sequent; (3) present an explicit way of generating the so-called invariant formula in induction-like rules. The saturated calculus for Horn-like sequents allows one: (1) to prove in an obvious way the completeness of a restricted linear temporal logic of the first order; (2) to construct a computer-aided proof system for this logic; (3) to prove the decidability of this logic for logically decidable Horn-like sequents. Bibliography: 15 titles.